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First question:

Let's say we have a hypothesis test:

${ H }_{ 0 }:u=100$ and ${ H }_{ 1 }:u\neq 100$.

The sample has a size of 10 and gives an average $u=103$ and a p-value = 0.08. The level of significance is 0.05.

I'm asked the following question (exam):

A) We can conclude that $u=100$

B) We cannot conclude that $u\neq100$

The 2 answers are rather similar, but not the same. I would say B) but I'm not so sure given what I've read.

The p-value here indicates that we cannot reject the null hypothesis, so we cannot accept H1 ?

Second question:

What does it mean exactly that a test is significant ?

Does it mean that we can reject the null hypothesis ?

Thanks in advance.

Regards,

XCoder
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3 Answers3

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A hypothesis test in the frequentist sense is a procedure by which one arrives at a decision about whether the data contains sufficient evidence to accept the alternative hypothesis. In other words, there are two choices: either you reject the null $H_0$, or the test is inconclusive.

The reason why you cannot ever "accept the null" under such a test is because the test statistic and the resulting $p$-value are calculated under the distributional assumption that the null is true. Therefore, a $p$-value that is not sufficiently small (i.e., smaller than the $\alpha$ level) is somewhat tautological: it essentially says that, assuming the data indeed is drawn from a distribution that follows the null hypothesis, the probability of seeing a sample as extreme as that you obtained is $p$. If $p$ is "large," that doesn't mean $H_0$ is true, because you assumed that it was true in order to get $p$ in the first place.

All that you can say if you fail to reject $H_0$ is that the data does not furnish enough evidence to suggest with a high degree of confidence that $H_0$ is false.

Another way to think of it is this: suppose I give you a coin and you wish to test if it is biased. If you toss it 10 times and observe 5 heads and 5 tails, that does not necessarily mean that it is in fact fair--you could have observed this result from a biased coin purely due to random chance. All you can say is that the result you obtained does not furnish strong evidence that the coin is biased.

heropup
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Aren't you also given either the population or sample deviation? If any of these are given, you assume the data is distributed a certain way, e.g., normally, with $\mu=100, \sigma=\sigma_0$. Knowing this, i.e., the (assumed) parameters of the population you are sampling from, then allows you to compute the probability of obtaining the value of 103 under this distribution (say , but not neccesarily, normal) $\mu=100, \sigma=\sigma_0$. If the probability of this value 103 is less than , say,(the significance level) 5% , then you reject ;otherwise you do not reject.

So rejecting a hypothesis at a significance level of k% given the assumption $\mu=\mu_0$, given some value s for either the sample deviation , when the sample mean is $\mu_2$ just means that the probability of obtaining a value of $\mu_2$ in a population with mean $\mu_0$ and deviation s is less than $k$%. Of course there are variants of this test for different parameters.

As above poster wrote, if the p-value-- the probability of observing the value you observed-- is less than the significance value/level, then you reject , but you reject at the given significance value/level. The explanation for what is going on is in the above paragraphs ; let me know if it was not clear.

user99680
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  • I don't have any info about the standard dev. This is just about the interpretation related to the p-value. Any clue ? – XCoder Aug 14 '14 at 20:30
  • Please see the bottom paragraph I just wrote and let me know if it answers your question. – user99680 Aug 14 '14 at 20:38
  • Sorry, I misread ;let me add something. – user99680 Aug 14 '14 at 20:39
  • No: the p-value > 0.05 here. – XCoder Aug 14 '14 at 20:45
  • Right, then you do not reject. But you do not conclude the mean is 100; the sample data may have been either unrepresentative or made up of outliers. You can only conclude that under assumptions (which you cannot guarantee) of data being representative (not outliers ) and randomly-selected, the claim is true. – user99680 Aug 14 '14 at 20:53
  • So you conclude answer B ? – XCoder Aug 14 '14 at 20:56
  • Phrasing is confusing. I would say we conclude/assume the mean is 100, but we cannot be 100% sure that this is the case. – user99680 Aug 14 '14 at 21:05
  • Yes it is confusing. Since the nb of observations is low (10), I think the right answer should be B ? – XCoder Aug 14 '14 at 21:12
  • But the number of observations is factored in the p-value already. – user99680 Aug 14 '14 at 21:14
  • oh ok. So we're now both to be confused. – XCoder Aug 14 '14 at 21:16
  • p-value normally indicates the plausibility that the null hypothesis is correct, not that it is correct, which leads me to think that the answer is B not A. Your opinion? – XCoder Aug 14 '14 at 21:19
  • It is always an issue of plausibility; you can never reach absolute certain, but it is more plausible/likely , given the sample data and the distribution, that the mean is actually 100. – user99680 Aug 14 '14 at 21:22
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A P-value can be reported more formally in terms of a fixed level α test. Here α is a number selected independently of the data, usually 0.05 or 0.01, more rarely 0.10. We reject the null hypothesis at level α if the P-value is smaller than α , otherwise we fail to reject the null hypothesis at level α.

Now figure out what you have to do? .

Satish Ramanathan
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  • Sorry, but I already know that. I know that I cannot reject the null hypothesis here. But does it mean that we can conclude that Ho=100 ??? This is the real question. Or can we just conclude that we cannot accept the alternative. – XCoder Aug 14 '14 at 20:46
  • We fail to reject the null hypothesis and there is weak evidence against the null hypothesis in favor of the alternative hypothesis and in general what you hope to conclude as a result of the experiment should be placed in the alternative hypothesis and hence I would say the answer is cannot conlude $u\ne 100$ – Satish Ramanathan Aug 14 '14 at 21:02